Abstract

We complete the finite-certificate task proposed for Family 119, without assuming either of its all-dimensional contraction claims. For the uniform cube $\{-1,1{\}}^2$ and independent symmetric noise of correlation $1/2$, we give the complete stochastic matrix, classify all sixteen Boolean functions, and prove the Hellinger inequality by exact algebra. Equality holds precisely for the four signed coordinates. Every other Boolean function has Hellinger gap greater than $27/1000$. For parity the gap lies strictly between $102/1000$ and $103/1000$; for the nonconstant biased orbit it lies strictly between $27/1000$ and $28/1000$. All comparisons reduce to displayed integer-square inequalities.

We prove the classical convexity argument that converts a Hellinger certificate into a Courtade–Kumar certificate. It yields, at the frozen instance, the sharp bound $1-h_2(1/4)$, the same equality cases, and a mutual-information gap greater than $1/50$ bit for every nonsigned-coordinate function. The rest of the manuscript freezes the distinct soft-information and Hellinger assertions, audits their proof interfaces, and separates arithmetic certification, sampling, implementation resources, and quantum recovery. The general Family 119 proofs remain unverified here. No quantum-channel, fidelity, gravitational, or spacetime conclusion is claimed.